Abstract

Using a new trick, the nonlinear equation is recast to a coupled system consisting of a linear equation and a nonlinear equation. For the latter, with a weight factor we split the nonlinear term into two parts on both sides of the equation. When the two-dimensional nonlinear system is linearized around the iteration point to be a linear system, we can easily solve it and develop a fast convergent iterative scheme to solve nonlinear equations. In order to further enhance the convergence speed, a linear term is added on both sides of the first linear equation, which results to a very powerful iterative scheme with parameter being analyzed by the eigenvalues. The new iterative scheme is proved to be absolutely convergent, and the number of iterations for convergence is estimated. The merits of the present iterative scheme are insensitive to the initial guess of the solution, convergent very fast, and without needing of the differentials of the function.

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